h8 - Stat 5102(Geyer Spring 2012 Homework Assignment 8 Due...

Stat 5102 (Geyer) Spring 2012 Homework Assignment 8 Due Wednesday, March 28, 2012 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 8-1. Show that each of the following is an exponential family. Identify the natural parameter and natural statistic. (a) The Poi( μ ) family of distributions. (b) The Exp( λ ) family of distributions. (c) The Gam( α,λ ) family of distributions with both parameters unknown. The natural parameter vector and natural statistic vector are both two-dimensional. 8-2. Suppose X is Poi( μ ) and the prior distribution for μ is Gam( α,λ ), where α and λ are hyperparameters. Find the posterior distribution for μ . 8-3. Suppose X 1 , ... , X n are IID Gam( α,λ ), where α is known and λ is unknown. Suppose the prior distribution for λ is Gam( α0 ,λ0 ), where α0 and λ0 are hyperparameters. Find the posterior distribution for

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